Maximal subgroups and PST-groups

A subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19-25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versiosn of Kaplan's results, which enables a better understanding of the relationships between these classes

Some classes of finite groups and mutually permutable products

This paper has been published in Journal of Algebra, 319(8):3343-3351 (2008). Copyright 2008 by Elsevier. http://dx.doi.org/10.1016/j.jalgebra.2007.12.001[EN] This paper is devoted to the study of mutually permutable products of finite groups. A factorised group G=AB is said to be a mutually permutable product of its factors A and B when each factor permutes with every subgroup of the other factor. We prove that mutually permutable products of Y-groups (groups satisfying a converse of Lagrange's theorem) and SC-groups (groups whose chief factors are simple) are SC-groups, by means of a local version. Next we show that the product of pairwise mutually permutable Y-groups is supersoluble. Finally, we give a local version of the result stating that when a mutually permutable product of two groups is a PST-group (that is, a group in which every subnormal subgroup permutes with all Sylow subgroups), then both factors are PST-groups.The second and the fourth authors have been supported by the Grant MTM2004-08219-C02-02 from MEC (Spain) and FEDER (European Union).Asaad, M.; Ballester Bolinches, A.; Beidleman, JC.; Esteban Romero, R. (2008). Some classes of finite groups and mutually permutable products. Journal of Algebra. 8(319). doi:10.1016/j.jalgebra.2007.12.001831

Prefactorized subgroups in pairwise mutually permutable products

The final publication is available at Springer via http://dx.doi.org/10.1007/s10231-012-0257-yWe continue here our study of pairwise mutually and pairwise totally permutable products. We are looking for subgroups of the product in which the given factorization induces a factorization of the subgroup. In the case of soluble groups, it is shown that a prefactorized Carter subgroup and a prefactorized system normalizer exist.Aless stringent property have F-residual, F-projector and F-normalizer for any saturated formation F including the supersoluble groups.The first and fourth authors have been supported by the grant MTM2010-19938-C03-01 from MICINN (Spain).Ballester-Bolinches, A.; Beidleman, J.; Heineken, H.; Pedraza Aguilera, MC. (2013). Prefactorized subgroups in pairwise mutually permutable products. Annali di Matematica Pura ed Applicata. 192(6):1043-1057. https://doi.org/10.1007/s10231-012-0257-yS104310571926Amberg B., Franciosi S., de Giovanni F.: Products of Groups. Clarendon Press, Oxford (1992)Ballester-Bolinches, A., Pedraza-Aguilera, M.C., Pérez-Ramos, M.D.: Totally and Mutually Permutable Products of Finite Groups, Groups St. Andrews 1997 in Bath I. London Math. Soc. Lecture Note Ser. 260, 65–68. Cambridge University Press, Cambridge (1999)Ballester-Bolinches A., Pedraza-Aguilera M.C., Pérez-Ramos M.D.: On finite products of totally permutable groups. Bull. Aust. Math. Soc. 53, 441–445 (1996)Ballester-Bolinches A., Pedraza-Aguilera M.C., Pérez-Ramos M.D.: Finite groups which are products of pairwise totally permutable subgroups. Proc. Edinb. Math. Soc. 41, 567–572 (1998)Ballester-Bolinches A., Beidleman J.C., Heineken H., Pedraza-Aguilera M.C.: On pairwise mutually permutable products. Forum Math. 21, 1081–1090 (2009)Ballester-Bolinches A., Beidleman J.C., Heineken H., Pedraza-Aguilera M.C.: Local classes and pairwise mutually permutable products of finite groups. Documenta Math. 15, 255–265 (2010)Beidleman J.C., Heineken H.: Mutually permutable subgroups and group classes. Arch. Math. 85, 18–30 (2005)Beidleman J.C., Heineken H.: Group classes and mutually permutable products. J. Algebra 297, 409–416 (2006)Carocca A.: p-supersolvability of factorized groups. Hokkaido Math. J. 21, 395–403 (1992)Carocca, A., Maier, R.: Theorems of Kegel-Wielandt Type Groups St. Andrews 1997 in Bath I. London Math. Soc. Lecture Note Ser. 260, 195–201. Cambridge University Press, Cambridge, (1999)Doerk K., Hawkes T.: Finite Soluble Groups. Walter De Gruyter, Berlin (1992)Maier R., Schmid P.: The embedding of quasinormal subgroups in finite groups. Math. Z. 131, 269–272 (1973

On some classes of supersoluble groups

This paper has been published in Journal of Algebra, 312(1):445-454 (2007). Copyright 2007 by Elsevier. http://dx.doi.org/10.1016/j.jalgebra.2006.07.035[EN] Finite groups G for which for every subgroup H and for all primes q dividing the index |G:H| there exists a subgroup K of G such that H is contained in K and |K:H|=q are called Y-groups. Groups in which subnormal subgroups permute with all Sylow subgroups are called PST-groups. In this paper a local version of the Y-property leading to a local characterisation of Y-groups, from which the classical characterisation emerges, is introduced. The relationship between PST-groups and Y-groups is also analysed.The first and the third authors have been supported by the Grant MTM2004-08219-C02-02 from MEC (Spain) and FEDER (European Union).http://dx.doi.org/10.1016/j.jalgebra.2006.07.035Ballester Bolinches, A.; Beidleman, JC.; Esteban Romero, R. (2007). On some classes of supersoluble groups. Journal of Algebra. 1(312). doi:10.1016/j.jalgebra.2006.07.035131

On non-commuting sets in finite soluble CC-groups

Lower bounds for the number of elements of the largest non-commuting set of a finite soluble group with a CC subgroup are considered in this paper

On second minimal subgroups of Sylow subgroups of finite groups

This paper has been published in Journal of Algebra, 342(2):134-146 (2011). Copyright 2011 by Elsevier. http://dx.doi.org/10.1016/j.jalgebra.2011.06.016A subgroup H of a finite group G is a partial CAP-subgroup of G if there is a chief series of G such that H either covers or avoids its chief factors. Partial cover and avoidance property has turned out to be very useful to clear up the group structure. In this paper, finite groups in which the second minimal subgroups of their Sylow p-subgroups, p a fixed prime, are partial CAP-subgroups are completely classified.The first and the second authors have been supported by the research grant MTM2010-19938-C03-01 from MICINN (Spain). Most of this research was carried out during a visit of the third author to the Departament d'Algebra, Universitat de Valencia, Burjassot, Valencia, Spain, during the academic year 2009-10.Ballester Bolinches, A.; Esteban Romero, R.; Li, Y. (2011). On second minimal subgroups of Sylow subgroups of finite groups. Journal of Algebra. 2(342). doi:10.1016/j.jalgebra.2011.06.016S234

On the p-length of some finite p-soluble groups

The main aim of this paper is to give structural information of a finite group of minimal order belonging to a subgroup-closed class of finite groups and whose p-length is greater than 1, p a prime number. Alternative proofs and improvements of recent results about the influence of minimal p-subgroups on the p-nilpotence and p-length of a finite group arise as consequences of our study

A question on partial CAP-subgroups of finite groups

This paper has been published in Science China Mathematics, 55(5):961-966 (2012). Copyright 2012 by Science China Press and Springer-Verlag. The final publication is available at www.springerlink.com. http://link.springer.com/article/10.1007/s11425-011-4356-9 http://dx.doi.org/10.1007/s11425-011-4356-9A subgroup H of a finite group G is a partial CAP-subgroup of G if there is a chief series of G such that H either covers or avoids every chief factor of the series. The structural impact of the partial cover and avoidance property of some distinguished subgroups of a group has been studied by many authors. However there are still some open questions which deserve an answer. The purpose of the present paper is to give a complete answer to one of these questions.This work was supported by MEC of Spain, FEDER of European Union (Grant No. MTM-2007-68010-C03-02), MICINN of Spain (Grant No. MTM-2010-19938-C03-01), National Natural Science Foundation of China (Grant No. 11171353/A010201) and Natural Science Fund of Guangdong (Grant No. S2011010004447). Part of this research was carried out during a visit of the third author to the Departament d'Algebra, Universitat de Valencia, Burjassot, Valencia, Spain, and the Institut Universitari de Matematica Pura i Aplicada, Universitat Politecnica de Valencia, Valencia, Spain, between September, 2009 and August, 2010. He is grateful to both institutions for their warm hospitality and, in particular, to the Universitat Politecnica de Valencia for the financial support given via its Programme of Support to Research and Development 2010.http://link.springer.com/article/10.1007/s11425-011-4356-9Ballester Bolinches, A.; Esteban Romero, R.; Li, Y. (2012). A question on partial CAP-subgroups of finite groups. Science China Mathematics. 5(55). doi:10.1007/s11425-011-4356-9S555Ballester-Bolinches A, Ezquerro L M. Classes of Finite Groups. In: Mathematics and its Applications, vol. 584. New York: Springer, 2006Ballester-Bolinches A, Ezquerro L M, Skiba A N. Local embeddings of some families of subgroups of finite group. Acta Math Sin Engl Ser, 2009, 25: 869–882Ballester-Bolinches A, Ezquerro L M, Skiba A N. On second maximal subgroups of Sylow subgroups of finite groups. J Pure Appl Algebra, 2011, 215: 705–714Doerk K, Hawkes T. Finite Soluble Groups. In: De Gruyter Expositions in Mathematics, vol. 4. Berlin-New York: Walter de Gruyter, 1992Ezquerro L M. A contribution to the theory of finite supersolvable groups. Rend Sem Mat Univ Padova, 1993, 89: 161–170Fan Y, Guo X Y, Shum K P. Remarks on two generalizations of normality of subgroups (in Chinese). Chinese Ann Math Ser A, 2006, 27: 169–176Guo X Y, Wang L L. On finite groups with some semi cover-avoiding subgroups. Acta Math Sin Engl Ser, 2007, 23: 1689–1696Huppert B. Endliche Gruppen I. In: Grund Math Wiss, vol. 134. Berlin-Heidelberg-New York: Springer-Verlag, 1967Huppert B, Blackburn N. Finite Groups III. In: Grund Math Wiss, vol. 243. Berlin: Springer-Verlag, 1982Li Y M. On cover-avoiding subgroups of Sylow subgroups of finite groups. Rend Sem Mat Univ Padova, 2010, 123: 249–258Li Y M, Miao L, Wang Y M. On semi cover-avoiding maximal subgroups of Sylow subgroups of finite groups. Comm Algebra, 2009, 37: 1160–116

On a theorem of Kang and Liu on factorised groups

[EN] Kang and Liu ['On supersolvability of factorized finite groups', Bull. Math. Sci. 3 (2013), 205-210] investigate the structure of finite groups that are products of two supersoluble groups. The goal of this note is to give a correct proof of their main theorem.The first author was supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad, Spain, and FEDER, European Union, and a project of Natural Science Foundation of Guangdong Province (No. 2015A030313791).Ballester-Bolinches, A.; Pedraza Aguilera, MC. (2018). On a theorem of Kang and Liu on factorised groups. Bulletin of the Australian Mathematical Society. 97(1):54-56. https://doi.org/10.1017/S0004972717000363S5456971Ezquerro, L. M., & Soler-Escrivà, X. (2003). On MutuallyM-Permutable Products of Finite Groups. Communications in Algebra, 31(4), 1949-1960. doi:10.1081/agb-120018515Kang, P., & Liu, Q. (2013). On supersolvability of fatorized finite groups. Bulletin of Mathematical Sciences, 3(2), 205-210. doi:10.1007/s13373-013-0032-4Ballester-Bolinches, A., Esteban-Romero, R., & Asaad, M. (2010). Products of Finite Groups. de Gruyter Expositions in Mathematics. doi:10.1515/9783110220612Ballester-Bolinches, A., Cossey, J., & Pedraza-Aguilera, M. C. (2001). ON PRODUCTS OF FINITE SUPERSOLUBLE GROUPS. Communications in Algebra, 29(7), 3145-3152. doi:10.1081/agb-501

On large orbits of supersoluble subgroups of linear groups

The research of this paper has been supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad, Spain, and FEDER, European Union, by the grant PGC2018-095140-B-I00 from the Ministerio de Ciencia, Innovacion y Universidades and the Agencia Estatal de Investigacion, Spain, and FEDER, European Union, and by the grant PROMETEO/2017/057 from the Generalitat, Valencian Community, Spain. The first author is supported by the predoctoral grant 201606890006 from the China Scholarship Council. The second author is supported by the grant 11401597 from the National Science Foundation of ChinMeng, H.; Ballester-Bolinches, A.; Esteban Romero, R. (2019). On large orbits of supersoluble subgroups of linear groups. Journal of the London Mathematical Society. 101(2):490-504. https://doi.org/10.1112/jlms.12266S4905041012Doerk, K., & Hawkes, T. O. (1992). Finite Soluble Groups. doi:10.1515/9783110870138Dolfi, S. (2008). Large orbits in coprime actions of solvable groups. Transactions of the American Mathematical Society, 360(01), 135-153. doi:10.1090/s0002-9947-07-04155-4Dolfi, S., & Jabara, E. (2007). Large character degrees of solvable groups with abelian Sylow 2-subgroups. Journal of Algebra, 313(2), 687-694. doi:10.1016/j.jalgebra.2006.12.004Espuelas, A. (1991). Large character degrees of groups of odd order. Illinois Journal of Mathematics, 35(3). doi:10.1215/ijm/1255987794The GAP group ‘GAP – groups algorithms and programming version 4.9.1’ 2018 http://www.gap‐system.org.Halasi, Z., & Maróti, A. (2015). The minimal base size for a -solvable linear group. Proceedings of the American Mathematical Society, 144(8), 3231-3242. doi:10.1090/proc/12974Huppert, B. (1967). Endliche Gruppen I. Grundlehren der mathematischen Wissenschaften. doi:10.1007/978-3-642-64981-3Keller, T. M., & Yang, Y. (2015). Abelian quotients and orbit sizes of solvable linear groups. Israel Journal of Mathematics, 211(1), 23-44. doi:10.1007/s11856-015-1259-4Manz, O., & Wolf, T. R. (1993). Representations of Solvable Groups. doi:10.1017/cbo9780511525971Meng, H., Ballester-Bolinches, A., & Esteban-Romero, R. (2019). On large orbits of subgroups of linear groups. Transactions of the American Mathematical Society, 372(4), 2589-2612. doi:10.1090/tran/7639Wolf, T. R. (1999). Large Orbits of Supersolvable Linear Groups. Journal of Algebra, 215(1), 235-247. doi:10.1006/jabr.1998.7730Yang, Y. (2009). Orbits of the actions of finite solvable groups. Journal of Algebra, 321(7), 2012-2021. doi:10.1016/j.jalgebra.2008.12.016Yang, Y. (2011). Large character degrees of solvable 3’-groups. Proceedings of the American Mathematical Society, 139(9), 3171-3173. doi:10.1090/s0002-9939-2011-10735-4Yang, Y. (2014). Large orbits of subgroups of solvable linear groups. Israel Journal of Mathematics, 199(1), 345-362. doi:10.1007/s11856-014-0002-